Concurrent Station Keeping, Attitude Control, and Momentum Management of Spacecraft

ABSTRACT

An operation of a spacecraft is controlled using an inner-loop control determining first control inputs for momentum exchange devices to control an orientation of the spacecraft and an outer-loop control determining second control inputs for thrusters of the spacecraft to concurrently control a pose of the spacecraft and a momentum stored by the momentum exchange devices of the spacecraft. The outer-loop control determines the second control inputs using a model of dynamics of the spacecraft including dynamics of the inner-loop control, such that the outer-loop control accounts for effects of actuation of the momentum exchange devices according to the first control inputs determined by the inner-loop control. The thrusters and the momentum exchange devices are controlled according to at least a portion of the first and the second control inputs.

FIELD OF THE INVENTION

This invention relates generally to controlling an operation of a spacecraft, and more particularly to concurrent station keeping, attitude control, and momentum management of spacecraft.

BACKGROUND OF THE INVENTION

A spacecraft in orbit is subject to various disturbance forces that affect its ability to maintain its station, i.e., desired orbit and position on the desired orbit. To counteract these forces, spacecraft are generally equipped with thrusters for station keeping maneuvers. Existing approaches to handle station keeping requirements use impulsive propulsion systems that are manually commanded from a ground control center.

In addition to orbital perturbations, spacecraft are disturbed by external torques that are generally absorbed by onboard momentum exchange devices, such as reaction wheels or control moment gyroscopes, allowing the spacecraft to maintain a desired orientation relative to the Earth or stars. To prevent saturation of the momentum exchange device and subsequent loss of the desired spacecraft attitude, the stored angular momentum is periodically unloaded via the onboard thrusters, which is also a manually commanded process from a ground control center.

The process of determining and commanding the onboard thrusters from a ground control center is manual, tedious and does not easily scale to the increasing number of spacecraft in particular orbits, e.g. geostationary orbit, and their tight station keeping windows as required, for example, for spacecraft co-location. Also, such a manual control results in an open-loop strategy, which is not able to automatically correct for errors introduced in the modeling or implementation of the desired station keeping and momentum management maneuvers, thus resulting in limited precision positioning and pointing of the spacecraft.

Generally station keeping and momentum unloading are achieved by a different set of thrusters, which is undesirable due to mass being a driving consideration in spacecraft design, and due to the increase in complexity and cost. Combined station keeping and momentum unloading problem using the same set of thrusters results in multiple objectives, and methods for coordinating such objectives in order to achieve them concurrently are challenging, see, e.g., method described in U.S. Pat. No. 8,282,043 that simplify the control by using maximum values available for torques and forces of the thrusters.

Furthermore, there are often restrictions on the placement of thrusters on a spacecraft so that antennas and solar panels may be deployed without the risk of thruster plume impingement. Restricting the placement of thrusters that are used for both station keeping and momentum unloading may mean that the thrusters would be unable to provide pure torques without also applying a net force on the satellite. Therefore, firing the thrusters may affect the spacecraft position and orientation (pose) as well as the stored momentum, creating a problem of concurrent station keeping, attitude control, and momentum management.

SUMMARY OF THE INVENTION

It is an object of some embodiments of an invention to provide a system and a method for concurrent control of an orbital position, orientation, and accumulated onboard momentum of a spacecraft using a single set of thrusters. It is another object of some embodiment to provide such a method that achieves the concurrent control using a model predictive control (MPC) over a receding horizon. It is further object of some embodiments to avoid control manually commanded from the ground and to provide an autonomous control that can be implemented in an onboard control system resulting in tighter and more accurate station keeping, attitude control, and momentum unloading.

Some embodiments of the invention are based on the realization that it is possible to use a single set of thrusters for concurrent station keeping, attitude control, and momentum unloading maneuvers by coordinating the requested thrust for both maneuvers amongst the available thrusters while respecting total thrust limitations. For example, the requirements of the station keeping, attitude control, and momentum management, such as a tight permissible station keeping window, tight orientation requirements, stringent constraints on available thrust, and coordination required between orbital control for the station keeping and attitude control for the momentum unloading, impose constraints on the states and the inputs of the controller.

It is an additional realization that a model predictive control (MPC) with a specifically defined model, a cost function, and constraints can be advantageous for generating fuel efficient maneuvers, which increase the effective life of the spacecraft. For example, electric thrusters have a much higher specific impulse than that of chemical thrusters, but require near continuous operation in order to provide a similar impulse. A multi-purpose MPC can address this problem using optimization of a cost function that includes components for controlling both the pose of the spacecraft and the momentum stored by the momentum exchange devices.

However, such a multi-purpose MPC needs to consider differences of station keeping and orientation dynamics. This is especially the concern for the spacecraft having thrusters located on a single face of the spacecraft, where forces and torques acting on the spacecraft are highly coupled. To that end, some embodiments consider effect that the MPC has on both the position and the orientation of the spacecraft.

Some embodiments of the invention are based on another realization that such a multi-purpose MPC needs to consider differences in the time-scales of station keeping (position control) and orientation dynamics (attitude control). For example, for some orbits, the time-scale of station keeping is significantly slower than the time-scale of attitude control, and a single MPC that solves both at the frequent state-sampling and control-inputs-computation required by the orientation dynamics while having long horizon of prediction required by station keeping is extremely computationally burdensome.

To that end, some embodiments of the invention employ a bi-level control scheme. For example, some embodiments control the spacecraft orientation using an inner loop control of the orientation of the spacecraft and use the outer-loop control, e.g., MPC, to control the pose of the spacecraft and a momentum stored by momentum exchange devices of the spacecraft. Because the inner and outer controls have different time scales due to the specifics of the spacecraft control, the inner loop control is decoupled from the outer-loop control, and the outer-loop control uses model including dynamics of the inner-loop control to account for the effects of the inner-loop control.

Accordingly, one embodiment of the invention discloses a method for controlling an operation of a spacecraft. The method includes determining, using an inner-loop control, first control inputs for momentum exchange devices to control an orientation of the spacecraft; determining, using an outer-loop control, second control inputs for thrusters of the spacecraft to concurrently control a pose of the spacecraft and a momentum stored by the momentum exchange devices of the spacecraft, wherein the outer-loop control determines the second control inputs using a model of dynamics of the spacecraft, wherein the model includes dynamics of the inner-loop control, such that the outer-loop control accounts for effects of actuation of the momentum exchange devices according to the first control inputs determined by the inner-loop control; and controlling the thrusters and the momentum exchange devices according to at least a portion of the first and the second control inputs. The steps of the method are performed by a processor.

Another embodiment discloses a control system for controlling an operation of a spacecraft according to a model of the spacecraft, comprising at least one processor for executing modules of the control system. The modules includes an inner-loop controller for determining first control inputs to momentum exchange devices for controlling an orientation of the spacecraft; an outer-loop controller for determining second control inputs to thrusters of the spacecraft for concurrently controlling a pose of the spacecraft and a momentum stored by the momentum exchange devices of the spacecraft, wherein the outer-loop controller determines the second control inputs using a model of dynamics of the spacecraft, wherein the model includes dynamics of the inner-loop control, such that the outer-loop controller accounts for effects of actuation of the momentum exchange devices according to the first control inputs determined by the inner-loop controller; and a mapper for controlling the thrusters and the momentum exchange devices according to at least a portion of the first and the second control inputs, wherein the controlling includes commanding to the momentum exchange devices to unload the stored momentum and commanding to the thrusters to generate a force and a torque to maintain or change the pose of the spacecraft and to compensate for a torque generated by the momentum exchange devices unloading the stored momentum.

Yet another embodiment discloses a spacecraft including a set of thrusters for changing a pose of the spacecraft, wherein the thrusters are located on a single face of the spacecraft; a set of momentum exchange devices for absorbing disturbance torques acting on the spacecraft; and a control system for controlling concurrently operations of the thrusters and the momentum exchange devices, the control system comprising at least one processor for executing modules of the control system.

The modules includes an inner-loop controller for determining first control inputs to momentum exchange devices for controlling an orientation of the spacecraft; an outer-loop controller for determining second control inputs to thrusters of the spacecraft for concurrently controlling a pose of the spacecraft and a momentum stored by the momentum exchange devices of the spacecraft, wherein the outer-loop controller determines the second control inputs using a model of dynamics of the spacecraft, wherein the model includes dynamics of the inner-loop control, such that the outer-loop controller accounts for effects of actuation of the momentum exchange devices according to the first control inputs determined by the inner-loop controller; and a mapper for controlling the thrusters and the momentum exchange devices according to at least a portion of the first and the second control inputs, wherein the controlling includes commanding to the momentum exchange devices to unload the stored momentum and commanding to the thrusters to generate a force and a torque to maintain or change the pose of the spacecraft and to compensate for a torque generated by the momentum exchange devices unloading the stored momentum.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A, 1B, and 1C are schematics of the problem formulation according to one embodiment of the invention;

FIG. 1D is a schematic of the gimbaling constraints on an individual thruster.

FIG. 2 is a block diagram of an inner-outer loop controller for controlling an operation of a spacecraft according to one embodiment of the invention;

FIG. 3 is a block diagram of a general structure of the controller of FIG. 1A according to one embodiment of the invention;

FIG. 4 is a block diagram of various modules of the controller according to one embodiment of the invention;

FIG. 5 is a block diagram of a method executed by the controller according to one embodiment of the invention; and

FIG. 6 is a schematic of the disturbance prediction problem according to one embodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

FIGS. 1A and 1B show a spacecraft 102 equipped with a plurality of actuators such as thrusters 150 and momentum exchange devices 151. Examples of the type of momentum exchange devices include reaction wheels (RWs) and gyroscopes. The spacecraft is a vehicle, vessel, or machine designed to fly in outer space whose operation changes quantities such as the position of the spacecraft, its velocities, and its attitude or orientation, in response to commands that are sent to the actuators. When commanded, the actuators impart forces on the spacecraft that increase or decrease the velocity of the spacecraft and thus cause the spacecraft to translate its position, and, when commanded, the actuators also impart torques on the spacecraft, which cause the spacecraft to rotate and thereby change its attitude or orientation. As used herein, the operation of the spacecraft is determined by the operation of the actuators that determine a motion of the spacecraft that changes such quantities.

The spacecraft flies in outer space along an open or closed orbital path 160 around, between, or near one or more gravitational bodies such as the Earth 161, moon, and/or other celestial planets, stars, asteroids, comets. Usually, a desired or target position 165 along the orbital path is given. A reference frame 170 is attached to the desired position, where the origin of the frame, i.e., the all zeros coordinates in that reference frame are the coordinates of the desired position at all times.

The spacecraft is subject to various disturbance forces 114. These disturbance forces can include forces that were not accounted for when determining the orbital path for the spacecraft. These disturbance forces act on the spacecraft to move the spacecraft away from the desired position on the orbital path. These forces can include, but are not limited to, gravitational attraction, radiation pressure, atmospheric drag, non-spherical central bodies, and leaking propellant. Thus, the spacecraft can be at a distance 167 away from the target position.

Because of the disturbance forces, it is not always possible to keep the spacecraft at the desired position along its orbit. As such, it is desired that the spacecraft instead remain within a window 166 with specified dimensions 604 around the desired position. To that end, the spacecraft is controlled to move along any path 606 that is contained within the window. In this example, the window 166 has a rectangular shape, but the shape of the window can vary for different embodiments.

The spacecraft is also often required to maintain a desired orientation. For example, a spacecraft-fixed reference frame 174 is required to be aligned with a desired reference frame such as an inertial reference frame 171 that is fixed relative to distant stars 172, or a reference frame 173 that is always oriented in a manner that points towards the Earth. However, depending on the shape of the spacecraft, different disturbance forces 114 can act non-uniformly on the spacecraft, thereby generating disturbance torques, which cause the spacecraft to rotate away from its desired orientation. In order to compensate for the disturbance torques, momentum exchange devices 151 such as reaction wheels are used to absorb the disturbance torques, thus allowing the spacecraft to maintain its desired orientation.

So that the momentum exchange devices do not saturate, and thereby lose the ability to compensate for disturbance torques, their stored momentum must be unloaded, e.g., by reducing spin rates of the reaction wheels. Unloading the momentum exchange devices imparts an undesired torque on the spacecraft. Such an undesired torque is also compensated for by the thrusters.

FIG. 1C shows the Euler Angles 175 between the spacecraft-fixed reference frame 174 and the desired reference frame 171. For example, some embodiments control the spacecraft such that the Euler Angles remain within limits 176 during the momentum unloading process. The thrusters 150 may be gimbaled in order to allow them to rotate a fixed amount from their nominal alignment with a thruster frame 181.

FIG. 1D shows a thrust force 180 generated by a thruster gimbaled from its nominal alignment. In this particular case, the thrust force is to be obtained by a thruster that is double gimbaled by two angles 185 that happen to be larger than the permitted gimbal angles 184. The gimbal angles 184 form the dimensions of a pyramid 183 and either due to the physical range or the permitted range of the gimbals, the thrust 180 must lie in the interior of the pyramid.

FIG. 2 shows a block diagram of a dual loop controller for controlling an operation of a spacecraft according to one embodiment of the invention. The motion of the spacecraft is affected by disturbance forces and torques 114. For example, the dual loop controller includes an inner-loop feedback control 108 controlling part of the operation of a spacecraft 102, for example, the orientation 174 of the spacecraft-fixed frame relative to a desired frame. The steps of the method can be implemented using a processor, e.g., the processor of the spacecraft and/or a remote processor

The inner loop control system receives information 110 about the spacecraft, which is a subset of the total information 106 about the spacecraft motion, from sensors, hardware, or software connected directly or remotely to the spacecraft. The information 106 includes a state of the spacecraft motion. The subset 110 is relevant for the inner-loop feedback control 108 and is used to generate commands 109 that in the case of orientation (attitude) control are commands to the momentum exchange devices 151. Also shown, is an outer loop control system 101 for controlling the operation of a spacecraft 102. The outer loop control system receives a target operation, e.g., a desired motion 103 for the spacecraft, such as a desired trajectory or a target point for some of the quantities, and controls the spacecraft via control inputs 107. The control inputs can include commands to change parameters of the operation of the spacecraft or can include actual values of the parameters such as voltages, pressures, torques, forces that affect the spacecraft.

The control inputs 107 together with the commands 109 form an input 104 to the spacecraft and induce a motion resulting in the generation of quantities 105 for the spacecraft. For example, the input 104 can be formed by a mapper 274. For example, the mapper 274 can combine the unmodified signals 107 and 109 to form the input 104. Additionally or alternatively, the mapper can determine commands to individual thrusters from a total commanded forces and torques for the spacecraft, so that the thrusters altogether impart the desired force and torques 107 to the spacecraft. The mapper 274 can pass the commands to the onboard momentum exchange devices without changing them along with the individual thruster commands as the current control input 104.

The outer-loop control system 101 also receives information 106 about the spacecraft motion. The outer-loop control system uses the state for the selection of the control inputs 107. The information 106 can include some or all of the motion quantities 105 and can also include additional information about the spacecraft. The quantities 105, the control inputs 107 or a combination thereof, can be requested to remain in some pre-defined ranges according to constraints 115 on the operation of the spacecraft.

It is an objective of some embodiments of the invention to determine the commands 107 to the thrusters 150 and the control inputs 109 to the momentum exchange devices 151 so that the spacecraft simultaneously stays within a box 166, has its Euler Angles 175 remain within limits 176, and unloads excess stored momentum. This is done by implementing an automatic outer-loop control system 101 that uses a model of the spacecraft 112 in conjunction with an inner-loop feedback control system 108. For example, some embodiments determine control inputs for controlling thrusters of the spacecraft using an optimization of a cost function over a receding horizon subject to constraints 115 on a pose of the spacecraft and inputs to the thrusters and generate appropriate control input commands 107. The pose of the spacecraft includes one or combination of an absolute or relative position and orientation of the spacecraft. In some embodiments, the cost function includes a component for controlling the pose of the spacecraft and a component for unloading a momentum stored by the momentum exchange devices.

For some embodiments, the inner-loop feedback control system 108 uses sensor measurements 110 to determine the current orientation of the spacecraft and sends commands 109 to actuators, such as momentum exchange devices 151, in order to reduce the error between the current orientation of the spacecraft and the target orientation of the spacecraft. In some embodiments, the inner-loop controller 108 applies proportional, integral, and derivative (PID) gains on the error between the current and the target orientations of the spacecraft as a feedback command 109 to the actuators. In other embodiments, the error between the current orientation and the target orientation of the spacecraft can be reduced using adaptive attitude controllers that generate a feedback command 109 based on internal states that estimate uncertain spacecraft parameters, e.g. moments and products of inertia. In some other embodiments, the inner-loop feedback control system 108 takes the form of an SO(3)-based attitude controller that encodes the error between the current spacecraft orientation and the target orientation of the spacecraft as a rotation matrix. The SO(3)-based attitude controller can provide almost-global asymptotic stability for attitude tracking problems, relatively easy to implement, and rejects the disturbance by providing infinite feedback gain at the disturbance frequencies. The target orientation of the spacecraft can be an inertially-fixed orientation and/or can be a specially changing orientation, e.g. the orientation that evolves in a manner that always points at the Earth.

In some embodiments, the outer-loop control system 101 achieves the control using a model predictive control (MPC) over a receding horizon. The MPC is based on an iterative, finite horizon optimization based on a model of the spacecraft including a component modeling the inner-loop feedback control, a set of objectives of the motion of the spacecraft, and constraints on the spacecraft propulsion system and motion, and has the ability to anticipate future events and consequently to take appropriate control actions. This is achieved by optimizing the operation of the spacecraft according the set of objectives, over a future finite time-horizon with prediction obtained according to the model of the spacecraft subject to constraints, and only implementing the control over the current timeslot. For example, the constraints can represent physical limitations of the spacecraft, safety limitations on the operation of the spacecraft, and performance limitations on a trajectory of the spacecraft. A control strategy for the spacecraft is admissible when the motion generated by the spacecraft for such a control strategy satisfies all the constraints. For example, at time t, the current state of the spacecraft is sampled and an admissible cost minimizing control strategy is determined for a relatively short time horizon in the future. Specifically, an online or real-time calculation determines a cost-minimizing control strategy until time t+T. After the first step of the control is implemented, the state is measured or estimated again and the calculations are repeated starting from the now current state, yielding a new control and new predicted state trajectory. The prediction horizon shifts forward, and for this reason MPC is also called receding horizon control.

FIG. 3 shows a general structure of the outer-loop control system 101 according to one embodiment of the invention. The control system 101 includes at least one processor 130 for executing modules of the controller. The processor 130 is operatively connected to a memory 120 for storing the spacecraft model 112 and the constraints 115. It is an objective of some embodiments of the invention to determine the control inputs 107 using a model of the spacecraft 112 subject to the constraints 115. The memory also can store the cost function 116. In one embodiment, the processor determines and/or updates at least one of the cost function, the constraints and the model during the control.

FIG. 4 shows a block diagram of various modules of the outer-loop control system 101 according to one embodiment of the invention. In some embodiments, the model of the spacecraft includes a nominal model 202 defining relationships among parameters of the model 112, such as the spacecraft orbital dynamics which governs translational motion of the spacecraft, and the spacecraft attitude dynamics and kinematics which governs attitude motion of the spacecraft. The model 112 also includes an inner-loop feedback control model 204 that allows the outer-loop to be able to predict the action taken by the inner-loop, and is based on the realization that the outer-loop control system 101 can command 107 the thrusters 150 to behave in a manner that causes the inner loop control system 108 to command 109 the momentum exchange devices 151 to unload their momentum.

The model 112 also includes a disturbance model 203 defining the disturbance forces 114 acting on the spacecraft. In some embodiments, the disturbance forces are determined as if the spacecraft is located at a predetermined position, e.g., the desired position 165, for different time steps of the control, i.e., regardless of the actual position of the spacecraft. Those embodiments are based on a realization that such an approximation simplifies the computational complexity of the disturbance without a significant reduction of the accuracy. The disturbance module 203 enables the MPC to exploit natural dynamics to compensate for the disturbance forces, so that fuel consumption can be reduced while satisfying motion objectives of the spacecraft.

Some of the spacecraft quantities need to remain in desired ranges defined by constraints 205 on the operation of the spacecraft. For example, such quantities can include a pose of the spacecraft including position constraints derived from the requirement to maintain the spacecraft within the window 166, and orientation constraints derived from the requirement to maintain the Euler Angles 175 within limits 181.

Some embodiments of the invention are based on the additional realization that constraints 206 on the control inputs are required in order to satisfy the operational limits of the spacecraft actuators, such as thrust magnitude limits. In some embodiments, the constraints 206 are used in combination with at least some constraints 205 for controlling the spacecraft.

In some embodiments, the control inputs 107 are determined based on an optimization of a cost function 209 subject to constraints on the operation of the spacecraft 205 and constraints on the control inputs 206. In some embodiments, the cost function includes a combination of multiple components, including a component 291 for the position of the spacecraft, a component 292 for the attitude of the spacecraft, a component 293 for the stored momentum, a component 294 for an objective of the operation of the spacecraft, and a component 295 for ensuring the stability of the operation of the spacecraft. By selecting different combinations of the components of the cost function 206, some embodiments determine a current cost function 209 adjusted for different objectives of the control.

For example, the component 291 for the position of the spacecraft penalizes a larger displacement 167 of the spacecraft from the desired position 165, so that the optimization of the cost function 209 results in control inputs that when applied to the spacecraft reduce the displacement 167 in order to help achieve the objective of remaining within the window 166.

The component 292 for the attitude of the spacecraft penalizes a larger magnitude of the Euler Angles 175 of the spacecraft between the spacecraft-fixed reference frame 174 and the desired reference frame, e.g. 171, so that the optimization of the cost function 209 results in control inputs that when applied to the spacecraft reduce the Euler Angles 175 in order to help achieve the objective of maintaining a desired orientation for the spacecraft. These resulting control inputs may be commands that induce the thrusters to fire in a manner that directly reduces the Euler Angles towards their objective, or may be commands that induce the thrusters to fire in a manner that influences the inner-loop control system to generate commands to the momentum exchange devices that help achieve the Euler Angle objective.

The component 293 for the stored momentum penalizes a larger magnitude of the stored momentum so that the optimization of the cost function 209 results in control inputs that when applied to the spacecraft unload the stored momentum, e.g., the high values of the reaction wheel spin rates are penalized, resulting in an optimization that produces control inputs to reduce the spin rates of the reaction wheels. As the outer-loop control system does not directly command the momentum exchange devices and cannot directly unload the stored momentum, this control input would cause the thrusters to fire in a manner that would influence the inner-loop control system to unload the stored momentum.

The component 294 for the objective of the operation of the spacecraft can, for example, include a penalty on the amount of fuel that the thrusters use in order that the optimization of the cost function 209 results in control inputs that use less fuel, or a penalty on a lower magnitude of the speed at which the spacecraft operates in order that the optimization of the cost function results in control inputs that cause the spacecraft to operate faster, i.e. achieve objectives in a shorter period of time.

The component 295 for the stability is determined such that the optimization of the cost function 209 results in control inputs that ensure the stability of the operation of the spacecraft. In one embodiment, where the desired orbit 160 is circular, the stability component of the cost function penalizes the position of the spacecraft at the end of the MPC horizon by using the solution to the Discrete Algebraic Riccati Equation (DARE). In other embodiments, the desired orbit is not circular. For example the desired orbit is elliptic, or otherwise non-circular and periodic. Then, the stability component penalizes the position of the spacecraft at the end of the MPC horizon by using the solution to the Periodic Differential Riccati Equation (PDRE). Note that the PDRE solution is not constant and thus the penalty for the current cost function 209 is selected to correspond to the PDRE solution at the time instant corresponding to the time at the end of the MPC horizon.

In some embodiments, each of the components 291-294 of the cost function 209 is weighted so that the optimization of the cost function produces control inputs that achieve the various individual component goals with priority corresponding to their relative weight.

For example, in one embodiment, the weights are selected so that the largest weight is given to the component 294 that penalizes the fuel that the thrusters use. As a result, this embodiment generates an operation of the spacecraft that prioritizes using the least amount of fuel possible at the expense of a larger average displacement 167. In a different embodiment, the largest weight is given to the component 291, which penalizes the displacement 167 from the desired position 165. As a result, this embodiment generates an operation of the spacecraft that prioritizes maintaining a small average displacement 167 at the expense of using more fuel. In some embodiments, the component 295 for stability has its weight defined according to the weight that generates a stabilizing control input.

The processor 130 of the control system 101 executes various modules of the control system including a control input module 208 for determining force commands 107 to the spacecraft thrusters during a current iteration by optimizing the current cost function 209. The control input module optimizes the current cost function using a current model 201 of the spacecraft subject to constraints 205 on the operation of the spacecraft and constraints 206 on the current control input.

In one embodiment, the optimization of the cost function 209 in the control input module 208 is formulated as a quadratic program (QP). Quadratic programs can be solved more quickly and efficiently in resource-constrained hardware such as spacecraft, which have limited onboard computational power. In order to take advantage of quadratic programs, a linear-quadratic MPC (LQ-MPC) formulation is applied.

For example, the control system also includes the current model module 201 for the linearization of the nominal model 202 at the desired target location 165 on the target orbit, the linearization of the inner-loop feedback control model 204, and determination of the disturbance forces at the desired target location 165. In some embodiments, the linearization is due to LQ-MPC making use of a linear prediction model. The module 201 determines the current model of the spacecraft for the current time instant and over the entire MPC horizon. The module 201 can also receive the current state of the spacecraft 106 to determine a state of the spacecraft relative to the linearization.

The control system also includes a cost function module 207 for determining the current cost function 209. For example, the cost function module updates the previous cost function based on change of the target operation of the spacecraft, e.g., a change in the desired motion 103, because different motions can necessitate different cost functions to have the quantities 105 for the spacecraft meet their desired objectives. Also, the cost function module can update the stability component 295 of the cost function if the desired orbit requires an updated weight based on the orbit. Because the steps of the control are performed iteratively, the current model and the current cost function become previous model and previous cost function for subsequent iteration. For example, the previous model, the previous cost function and the previous control input are determined at a previous iteration as the current model, the current cost function and the current control input.

FIG. 5 shows a block diagram of a method executed by the modules of the outer-loop control system 101. The method controls iteratively the operation of the spacecraft with control inputs determined using the model of the spacecraft based on an optimization of a cost function. The method determines 210 a current state of the spacecraft resulted from the controlling with a previous control input determined for a previous iteration by optimizing a previous cost function using a previous model of the spacecraft. The current spacecraft state can be determined using hardware, software, or communication with the ground, e.g. GPS, relative range measurements, star trackers, horizon sensors.

In one embodiment, prior to determining the current control input, the method updates the model 230. For example, the model update includes linearization of the spacecraft model 112 at the desired target location 165 on the target orbit for the current time instant and over a future prediction horizon. The model update also computes the predicted disturbance forces 203 over the same horizon at the target location and combines it with the dynamics prediction model to form an overall prediction. Finally, in some embodiment, the model update 230 can also update the stability component 295 of the cost function to the correct value for the current time instant and over a future prediction horizon.

Next, the method determines 280 a current control input for controlling the spacecraft at the current iteration using the current model and the current cost function. For example, the method uses the updated current cost function and current spacecraft model to determine 240 a sequence of future inputs of thruster forces from current time instant for a fixed amount of time in the future, long at least as to obtain a new spacecraft state measurement, such that the predicted future spacecraft states and inputs satisfy the constraints on the operation of the spacecraft and constraints on the control inputs. The first part of the input sequence, for duration equal to the amount of time needed to obtain a new measurement of the state of the spacecraft, is selected 250 and is applied 260 as current control input to the spacecraft. Based on the current state of the spacecraft, current model of the spacecraft, and current control input to the spacecraft, the next state of the spacecraft is determined, and the controller waits 270 until a new state measurement is received.

Equations Used for Computing the Commands to the Thrusters

In one embodiment of the invention, the spacecraft model 112 is determined for a nadir-pointing spacecraft in geostationary Earth orbit (GEO) equipped with four gimbaled electric thrusters 150 and three axisymmetric reaction wheels 151 attached to a rigid bus in an orthogonal and mass balanced configuration. A bus-fixed frame 174 is defined for the spacecraft, and an inertial frame 171 is specified for determining the attitude of the spacecraft. The spacecraft equations of motion are given by

$\begin{matrix} {{{{\overset{¨}{r}}_{g}^{c\; \omega} + {\mu \frac{r_{g}^{c\; \omega}}{{r_{g}^{c\; \omega}}^{3}}}} = {a_{g}^{p} + {\frac{1}{m_{\mathcal{B}}}f_{g}^{thrust}}}},{{{J_{p}^{\mathcal{B}\; c}{\overset{.}{\omega}}_{p}^{pg}} + {\omega_{p}^{{pg}^{x}}\left( {{J_{p}^{\mathcal{B}\; c}\omega_{p}^{pg}} + {J_{s}\overset{.}{\gamma}}} \right)} + {J_{s}\eta}} = {\tau_{p}^{p} + \tau_{p}^{thrust}}},{C_{pg} = {{- \omega_{p}^{{pg}^{x}}}C_{pg}}},{\overset{¨}{\gamma} = \eta},} & (1) \end{matrix}$

where r_(g) ^(cw) is the position of the satellite, q^(pg) is an attitude parametrization of C_(pg) with is the rotation matrix of the bus frame relative to the inertial frame, and γ is a column matrix containing the angle of rotation of each reaction wheel. The vector ω_(p) ^(pg) is the angular velocity of the bus frame with respect to the inertial frame resolved in the bus frame. The matrix J_(p) ^(Bc) is the moment of inertia of the satellite B relative to its center of mass, resolved in the bus frame. The reaction wheel array has a moment of inertia J_(s) and the wheels are controlled with an acceleration η. The term a_(g) ^(p) represents the external perturbations on the satellite due to Earth's non spherical gravitational field, solar and lunar gravitational attraction, solar radiation pressure (SRP), and are defined below in (4). The term τ_(p) ^(p) represents the perturbation torques due to the solar radiation pressure, which assumes total absorption and is given by

$\tau_{p}^{p} = {{- {c_{p}\left\lbrack {r_{p}^{p_{i}c}{\sum\limits_{i = 1}^{N_{s}}\; {n_{p}^{i^{T}}{\hat{r}}_{p}^{sc}}}} \right\rbrack}^{x}}{\hat{r}}_{p}^{sc}}$

where c_(p) is the effective SRP near the Earth, {circumflex over (r)}_(p) ^(sc) is a unit vector pointing towards the sun from the center of mass of the spacecraft, p_(i) is the center of pressure of one of the six sides of the satellite, r_(p) ^(p) ^(i) ^(c) is the position of the center of pressure of the i-th panel relative to the center of mass of the satellite, and n_(p) ^(i) is the normal vector of the i-th panel resolved in the bus frame. The value N_(s) is the number sun-facing panels. Only sun-facing panels contribute to the torque. The gimbaled electric thrusters 150 produce forces that provide a net force on the spacecraft given by

$\begin{matrix} {f_{g}^{thrust} = {{\sum\limits_{i = 1}^{4}\; f_{g}^{t_{i}}} = {C_{pg}^{T}{\sum\limits_{i = 1}^{4}{C_{ip}^{T}{f_{g}^{t_{i}}.}}}}}} & (3) \end{matrix}$

They also produce a net torque on the spacecraft given by

$\begin{matrix} {\tau_{p}^{thrust} = {\sum\limits_{i = 1}^{4}{r_{p}^{t_{i}c^{x}}C_{ip}^{T}{f_{i}^{t_{i}}.}}}} & (4) \end{matrix}$

In one embodiment, the inner-loop control system 108 is an SO(3)-based attitude controller, where the feedback law controlling the reaction wheel array in (1) is given by

η=−J _(s) ⁻¹(ν₁+ν₂+ν₃)  (11)

where

-   -   ν₁=ω_(p) ^(pg) ^(x) (J_(p) ^(B) ^(c) ω_(p) ^(pg)+J_(s){dot over         (γ)})−J_(p) ^(B) ^(c) (K₁{dot over (S)}+{tilde over         (ω)}^(x)ω_(p) ^(pg)),     -   ν₂=−{circumflex over (τ)}_(dist),     -   ν₃=−K_(υ)({tilde over (ω)}+K₁S)−K_(p)S,         and

{circumflex over ({dot over (d)})}=A _(d) {circumflex over (d)}+B _(d)(ω_(p) ^(pd) +K ₁ S),

{circumflex over (τ)}_(dist) =C _(d) {circumflex over (d)},  (12)

This SO(3) inner-loop attitude controller makes use of the following quantities

S=−

_(a)(C _(pd))^(v),

S=

_(a)(ω_(p) ^(pd) ^(x) C _(pd))^(v),

ω_(p) ^(pd)=ω_(p) ^(pg) −C _(pd)ω_(d) ^(dg),

where

_(a) is a skew-symmetric projection operator, (•)^(v): so(3)→

³: is the uncross operator un-mapping a skew-symmetric matrix into a three dimensional vector, and K₁, K_(ν), and K_(p) are gains.

In other embodiments, the equations in (1) are substituted for equations that govern a spacecraft in other orbits and with other momentum exchange devices other than reaction wheels.

In one embodiment, the model (1) is linearized to form a current prediction model 201. The attitude-error rotation matrix {tilde over (R)}=R^(T)R_(d) is parameterized using the set of 3-2-1 Euler angles (ψ, θ, φ) as {tilde over (R)}=C₁(φ)C₂(θ)C₃(ψ), where R_(d) is the desired attitude trajectory, and C₁, C₂, and C₃ are elementary rotations about the x, y, and z-axes by ψ, θ, and φ, respectively. The linearization of the attitude dynamics and kinematics about an equilibrium spin with an angular rate corresponding to the mean motion n of the orbit, along with the linearization of the motion of the spacecraft for small maneuvers around a nominal circular orbit, is given by

${\delta \overset{¨}{r}} = {{{- \Omega}\; \delta \; r} - {2\; \omega_{0}^{x}\delta \overset{.}{r}} + a_{h}^{p} + {\frac{1}{m_{\mathcal{B}}}C_{dh}^{T}{\sum\limits_{i = 1}^{4}{C_{ip}^{T}f_{i}^{ti}}}}}$ δ θ̂ = −ω₀^(x)δ θ + δ ω ${\delta \; \overset{.}{\omega}} = {{{- {J_{p}^{\mathcal{B}\; c^{- 1}}\left( {{\omega_{0}^{x}J_{p}^{\mathcal{B}\; c}} - \left( {J_{p}^{\mathcal{B}\; c}\omega_{0}} \right)^{x}} \right)}}{\delta\omega}} - {J_{p}^{\mathcal{B}\; c^{- 1}}\omega_{0}^{x}J_{s}\delta \overset{.}{\overset{.}{\gamma}}} - {J^{\mathcal{B}\; c_{p}^{- 1}}J_{s}\eta} + {\sum\limits_{i = 1}^{4}{r_{p}^{t_{i}c^{x}}C_{ip}^{T}{f_{i}^{ti}.}}}}$

where δx, δy and δz are the components of the relative position vector δr of the spacecraft relative to the nominal location 165, Ω=diag{−3n²,0,n²}, n=√{square root over (μ/R₀ ³)} is the mean motion of the nominal orbit, δw=[δω₁, δω₂, δω₃] is the relative angular velocity components of the spacecraft, and δθ=[δφ, δθ, δψ] is the relative Euler angles of the spacecraft. That is, they are quantities that represent the error from the desired spacecraft angular velocity components and desired Euler angles.

For embodiments that utilize an inner-loop attitude control, the linearization of (1) along with the inner-loop control law (11), is used as the inner-loop feedback control model 204 as part of the prediction based on the model 112. This gives the changed relative angular velocity equation

${\delta \overset{.}{\omega}} = {\tau_{p}^{thrust} + {\underset{K_{d\; 2}}{\underset{}{\left\lbrack {{- K_{1}} + \omega_{0}^{x} - {J_{p}^{{\mathcal{B}\; c} - 1}K_{\upsilon}}} \right\rbrack}}\delta \; \omega} - {J_{p}^{{\mathcal{B}\; c} - 1}C_{d}\hat{d}} + {\underset{K_{p\; 2}}{\underset{}{\left\lbrack {{K_{1}\omega_{0}^{x}} - {\omega_{0}^{x}\omega_{0}^{x}} + {J_{p}^{{\mathcal{B}\; c} - 1}\left( {{K_{\upsilon}\omega_{0}^{x}} - {K_{\upsilon}K_{1}} - K_{p}} \right)}} \right\rbrack}}\delta \; \theta_{1}}}$

and the additional linearized equations based on (12) given by

{circumflex over ({dot over (d)})}=A _(d) {circumflex over (d)}+B _(d) δω+B _(d)(K ₁−ω₀ ^(x))δθ,

{circumflex over (τ)}_(dist) =C _(d) {circumflex over (d)}.

For embodiments in which a spacecraft is in GEO, the main perturbation accelerations are due to solar and lunar gravitational attraction, solar radiation pressure, and the anisotropic geopotential, that is, Earth's non-spherical gravitational field. Analytic expressions for these perturbation forces per unit mass, i.e., the disturbance accelerations, are given, respectively, by

$\begin{matrix} {{{\overset{\rightharpoonup}{a}}_{sun} = {\mu_{sun}\left( {\frac{{\overset{\rightharpoonup}{r}}_{{sun}/{sc}}}{r_{{sun}/{sc}}^{3}} - \frac{{\overset{\rightharpoonup}{r}}_{{sun}/{earth}}}{r_{{sun}/{earth}}^{3}}} \right)}},{{\overset{\rightharpoonup}{a}}_{moon} = {\mu_{moon}\left( {\frac{{\overset{\rightharpoonup}{r}}_{{moon}/{sc}}}{r_{{moon}/{sc}}^{3}} - \frac{{\overset{\rightharpoonup}{r}}_{{moon}/{earth}}}{r_{{moon}/{earth}}^{3}}} \right)}},{{\overset{\rightharpoonup}{a}}_{srp} = {C_{srp}\frac{S\left( {1 + c_{refl}} \right)}{2\; m}\frac{{\overset{\rightharpoonup}{r}}_{{sc}/{sun}}}{r_{{sc}/{sun}}}}},{{\overset{\rightharpoonup}{a}}_{J_{2}} = {\frac{3\mu \; J_{2}\rho_{E}^{2}}{2r^{5}}\left( {{\left( {{5\frac{\left( {\overset{\rightharpoonup}{r} \cdot {\hat{k}}_{E}} \right)}{r^{2}}} - 1} \right)\overset{\rightharpoonup}{r}} - {2\left( {\overset{\rightharpoonup}{r} \cdot {\hat{k}}_{E}} \right){\hat{k}}_{E}}} \right)}},} & (4) \end{matrix}$

where

denotes a coordinate-free (unresolved) vector, μsun and μmoon are the gravitational constants of the sun and moon, C_(srp) is the solar radiation pressure constant, S is the solar-facing surface area, c_(refl) is the surface reflectance, ρ_(E) is Earth's equatorial radius, {circumflex over (k)}_(E) is the z-axis unit vector of the Earth-centered inertial frame, and J₂ is the dominant coefficient in the considered geopotential perturbation model, where additional higher order terms are ignored. The sum of the individual disturbance accelerations in (4) yields the total disturbance acceleration considered in (1).

In some embodiments, a state-space model is given by

{dot over (x)}(t)=A _(c) x(t)+B _(c) u(t),  (5)

where

x=[δr ^(T) δ{dot over (r)} ^(T)δθ^(T) ^(˜) δω^(T)δ{dot over (γ)}^(T) {circumflex over (d)} ^(T)]^(T)  (6)

u=[f ₁ ^(t) ¹ f ₂ ^(t) ² f ₃ ^(t) ³ f ₄ ^(t) ⁴ ]^(T)  (7)

In order to be used as a prediction model in the MPC policy, (5) is discretized with a sampling period of ΔT sec which yields

x _(k+1) Ax _(k) +Bu _(k),  (8)

where x_(k) is the state at time step kεZ⁺, u_(k) is the control vector at the time step kεZ⁺, and A=exp(A_(c)ΔT), B=∫₀ ^(ΔT)exp(A_(c)(ΔT−τ))dτB_(c) are the discretized matrices obtained based on the continuous-time system realization (A_(c), B_(c)) in (5).

Estimation of the Disturbances Acting on the Spacecraft

In some embodiments, the model (8) is augmented with a prediction 203 model of the disturbance accelerations (4), obtaining

x _(k+1) =Ax _(k) +Bu _(k) +O _(H/E) a _(p,k),  (9)

where a_(p,k) is the total disturbance acceleration predicted at time step k based on propagation of the desired position 165, and O_(H/E) is the rotation matrix that transforms the components of a_(p,k) from the inertial frame 171 into the components of the same acceleration in the desired reference frame 170.

The desired position 165 for disturbance-acceleration prediction is used in (9) due to the nonlinearity of the analytical expressions in (4).

FIG. 6 shows the spacecraft 102 displaced from its desired position 165 at time step k=0. Because the desired positions 601, 602, and 603 on the nominal orbit 160 at time steps k=1, k=2, . . . , k=N are known in advance, a_(p,k) can be predicted based on the analytical expressions (4) at time steps k=1, k=2, . . . , k=N from the disturbance forces 604, 605, and 606. As the spacecraft position is to be constrained in a tight window 166, the difference in the disturbance accelerations at the desired positions 601, 602, and 603 and at the true satellite position 607, which is unknown in advance, is negligible. Accordingly, some embodiments determine the disturbance forces as if the spacecraft is located at the target position for the entire period of the receding horizon.

Constraints on Inputs to Thrusters

In some embodiments, constraints 205 on the operation of the spacecraft are imposed, at least in part, by δy and δz, corresponding to a station keeping window 166 using the relations

|δy|≦r ₀ tan(Δ_(1,max)),  (10a)

|δz|≦r ₀ tan(Δ_(2,max)),  (10b)

where λ_(1,max) is the maximum tolerable longitude error, and λ_(2,max) is the maximum tolerable latitude error.

In some embodiments of the invention, the thrusters 150 may be gimbaled in order to allow them to rotate a fixed amount from their nominal alignment with a thruster frame 181. Individual thrusters, however, have limited gimbal range of motion and are therefore constrained to lie in the interior of four planes that form a pyramid 183. This constraint is given by

D _(i) f _(i) ^(t) ^(i) ≧0,  (13)

where each of the four rows of D_(i) contains a normal vector describing a plane.

In some embodiments, the relative Euler angles (δφ, δθ, δψ) are constrained to be within a small tolerance,

|δφ|≦δφ_(max), |δθ|≦δθ_(max), |δψ|≦δψ_(max),  (14)

in order maintain the spacecraft orientation, even while unloading excess stored momentum.

Cost Function Objectives

In some embodiments, the current cost function 209 is composed of costs associated with various objectives, e.g. an objective J₁ that quantifies displacement from the nominal orbital position, an objective J₂ that quantifies the error in the Euler angles and penalizes the spacecraft angular velocity components, an objective J₃ that penalizes usage of the thrusters to generate forces and torques, and an objective J₄ that penalizes the reaction wheel momentum. In some embodiments, these costs J₁-J₄ are given by

J ₁=(δz)²+(δy)²+(δz),²

J ₂=(δφ)2+(δθ)²+(δψ)²+(δ/1)²+(δ/2)²+(δ/3),²

J ₃=(F _(x))²+(F _(y))²+(F _(z))²+(→₁)²+(→₂)²+(→₃),²

J ₄=(□₁)²+(□₂)²+(□₃).²

Each objective J₁-J₄ is multiplied by a weight w; and combined into a total cost function J_(tot),

J _(tot)=♡_(i=1, . . . ,4) w _(i) J _(i).  (18)

The weight w_(i) assigned to each objective determines its relative importance. The larger the weight assigned to given objective, the more that objective takes precedence when the cost function is optimized.

Based on (6) and (7), J_(tot) can be written for the state-space formulation as

J _(tot) =x ^(T) Qx+u ^(T) Ru,  (19)

where Q and R are symmetric positive definite weighting matrices that encode the weights w_(i) assigned to each objective and may further modify or add additional weights such as cross-weights that are not evident from the component formulation (18).

Stability Objective of the Cost Function

In some embodiments, where the desired orbit is not circular, for example elliptic, or otherwise non-circular and periodic, then the model 201 of the spacecraft motion about that orbit may be linear and time-varying. In such embodiments, the component 295 of the cost function 209 for the stability is determined based on the solution to the Periodic Difference Riccati Equation (PDRE)

P _(k) =Q _(k) +A _(k) ^(T) P _(k+1) A _(k) −A _(k) ^(T) P _(k+1) B _(k)(R _(k) +B _(k) ^(T) P _(k+1) B _(k))⁻¹ B _(k) ^(T) P _(k+1) A _(k)  (15)

where A_(k), B_(k) are the matrices of the model 201 at time step k, and P_(k), Q_(k), and R_(k), are symmetric positive definite weighting matrices. The matrices Q_(k) and R_(k) are taken to be the same as the weighting matrices in (19).

For embodiments where the linearization is time-invariant, such as motion around a nominal circular orbit, e.g. GEO, the component 295 for the stability is determined based on the solution to the Discrete Algebraic Riccati Equation (DARE)

P=Q+A ^(T) PA−A ^(T) PB(R+B ^(T) PB)⁻¹ B ^(T) PA  (16)

where A, B are the matrices of the model in (8), and P, Q, and R, are symmetric positive definite weighting matrices. As above, the matrices Q and R are taken to be the same as the weighting matrices in (19).

Control Input Computation

In some embodiments, the control input module 208 takes the form of a finite horizon numerical optimization problem,

$\begin{matrix} \begin{matrix} \min\limits_{U} & {{{x_{N}^{T}P_{N}x_{N}} + {\sum\limits_{k = 1}^{N - 1}\; {x_{k}^{T}Q_{k}x_{k}}} + {u_{k}^{T}R_{k}u_{k}}},} \\ {s.t.} & {{x_{k + 1} = {{A_{k}x_{k}} + {B_{k}u_{k}} + {_{{H/E},k}a_{p,k}}}},} \\ \; & {x_{0} = {x(t)}} \\ \; & {{T_{\min} \leq {Du}_{k} \leq T_{\max}},} \\ \; & {{{\delta \; y_{\min}} \leq {\delta \; y} \leq {\delta \; y_{\max}}},} \\ \; & {{{\delta \; z_{\min}} \leq {\delta \; z} \leq {\delta \; z_{\max}}},} \\ \; & {{{\delta \; \varphi_{\min}} \leq {\delta \; \varphi_{k}} \leq {\delta \; \varphi_{\max}}},} \\ \; & {{{\delta \; \theta_{\min}} \leq {\delta \; \theta_{k}} \leq {\delta \; \theta_{\max}}},} \\ \; & {{{\delta \; \psi_{\min}} \leq {\delta \; \psi_{k}} \leq {\delta \; \psi_{\max}}},} \end{matrix} & (17) \end{matrix}$

which is formed from the current cost function 209, the current linearized spacecraft model 201 that predicts the evolution of the state over the horizon using (9), and the spacecraft constraints 206 using (10), (13), and (14), where P_(N), Q_(k), R_(k) are the matrices given in (15), D is as in (13), and x(t) is the state at the current time step. The problem (17) is solved using a numerical solver, which finds the input sequence U=[u₁ . . . u_(N)]^(T) that minimizes the current cost function subject to the problem constraints.

The first input u₁ in the input sequence is considered as the output 107 of the input computation 208. The input u₁ is combined with the output of the inner-loop feedback control 109 which constructs the commands 104 to the thrusters and the momentum exchange devices. At the next time step, t+1 the model and cost function are updated, the state is updated, and the numerical optimization problem is solved again.

If the orbit is such that the spacecraft model 201 is time-invariant, then A₁=A₂= . . . =A, B₁=B₂= . . . =B, and P_(N), Q_(k), and R_(k) in (17) are given by the matrices P, Q, and R in (16). The inclusion of P_(N) or P in the cost function of (17) ensures local stability of the target position, as near the origin, where constraints are inactive, and in the absence of disturbance prediction, the solution of (17) is equivalent to that of either a Periodic-LQR or LQR controller.

The above-described embodiments of the present invention can be implemented in any of numerous ways. For example, the embodiments may be implemented using hardware, software or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers. Such processors may be implemented as integrated circuits, with one or more processors in an integrated circuit component. Though, a processor may be implemented using circuitry in any suitable format.

Further, it should be appreciated that a computer may be embodied in any of a number of forms, such as a rack-mounted computer, a desktop computer, a laptop computer, minicomputer, or a tablet computer. Such computers may be interconnected by one or more networks in any suitable form, including as a local area network or a wide area network, such as an enterprise network or the Internet. Such networks may be based on any suitable technology and may operate according to any suitable protocol and may include wireless networks, wired networks or fiber optic networks.

Also, the various methods or processes outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages and/or programming or scripting tools.

Also, the embodiments of the invention may be embodied as a method, of which an example has been provided. The steps performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

Although the invention has been described by way of examples of preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the invention. Therefore, it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention. 

1. A method for controlling an operation of a spacecraft, comprising: determining, using an inner-loop control, first control inputs for momentum exchange devices to control an orientation of the spacecraft; determining, using an outer-loop control, second control inputs for thrusters of the spacecraft to concurrently control a pose of the spacecraft and a momentum stored by the momentum exchange devices of the spacecraft, wherein the outer-loop control determines the second control inputs using a model of dynamics of the spacecraft, wherein the model includes dynamics of the inner-loop control, such that the outer-loop control accounts for effects of actuation of the momentum exchange devices according to the first control inputs determined by the inner-loop control; and controlling the thrusters and the momentum exchange devices according to at least a portion of the first and the second control inputs, wherein steps of the method are performed by a processor.
 2. The method of claim 1, wherein the outer-loop control is a model predictive control (MPC) that optimizes a cost function over a receding horizon subject to constraints on the pose of the spacecraft and constraints on inputs to the thrusters, wherein the cost function includes components for controlling the pose of the spacecraft and the momentum stored by the momentum exchange devices.
 3. The method of claim 2, wherein the constraints on the inputs to the thrusters define a range of rotations of the thrusters located on a single face of the spacecraft.
 4. The method of claim 3, wherein the range of rotations forces thrusts of the thrusters to lie within an interior of a pyramid.
 5. The method of claim 2, wherein the constraints on the pose of the spacecraft include a position constraint maintaining a position of the spacecraft within a predetermined window and an orientation constraint maintaining Euler Angles of the spacecraft within a predetermined limit, and wherein the constraints on the inputs to the thrusters guarantees an ability of the thrusters to jointly generate a force for controlling the pose of the spacecraft and a torque for unloading the momentum stored by the momentum exchange devices of the spacecraft.
 6. The method of claim 2, further comprising: determining the cost function as a combination of multiple components including a component for a position of the spacecraft penalizing a displacement of the spacecraft from a desired position, a component for an attitude of the spacecraft penalizing larger values of Euler Angles of the spacecraft, a component for the stored momentum penalizing a larger value of a magnitude of the stored momentum, a component for an objective of the operation of the spacecraft, and a component ensuring a stability of the operation of the spacecraft.
 7. The method of claim 6, further comprising: weighting each of the components of the cost function, such that the optimization of the cost function produces control inputs that achieve goals of each individual component with priority corresponding to their relative weight.
 8. The method of claim 7, wherein the control inputs are determined iteratively, and wherein at least one iteration comprises: updating one or combination of the components of the cost function and weights of the components of the cost function based on a change of a desired operation of the spacecraft.
 9. The method of claim 2, wherein model of dynamics of the spacecraft includes a nominal model defining relationships among parameters of the model and a disturbance model defining disturbance forces acting on the spacecraft, further comprising: performing a linearization of the nominal model as if the spacecraft is located at a target position for the entire period of the receding horizon; and determining the disturbance forces as if the spacecraft is located at the target position for the entire period of the receding horizon.
 10. The method of claim 1, wherein the controlling includes commanding the momentum exchange devices to unload the stored momentum and commanding to the thrusters to generate a force and a torque to maintain or change the pose of the spacecraft and compensate for a torque generated by the momentum exchange devices unloading the stored momentum.
 11. The method of claim 10, wherein the second control inputs determined by the outer-loop control specify gimbal angles and propulsion thrusts of the thrusters, and wherein the first control inputs determined by the inner-loop control specify control inputs [what kind of?] to the momentum exchange devices.
 12. The method of claim 1, wherein the inner-loop control reduces an error between the orientation of the spacecraft and a target orientation of the spacecraft.
 13. The method of claim 1, the second control inputs force the inner loop control to reduce speed of the momentum exchange devices.
 14. A control system for controlling an operation of a spacecraft according to a model of the spacecraft, comprising at least one processor for executing modules of the control system, the modules comprising: an inner-loop controller for determining first control inputs to momentum exchange devices for controlling an orientation of the spacecraft; an outer-loop controller for determining second control inputs to thrusters of the spacecraft for concurrently controlling a pose of the spacecraft and a momentum stored by the momentum exchange devices of the spacecraft, wherein the outer-loop controller determines the second control inputs using a model of dynamics of the spacecraft, wherein the model includes dynamics of the inner-loop control, such that the outer-loop controller accounts for effects of actuation of the momentum exchange devices according to the first control inputs determined by the inner-loop controller; and a mapper for controlling the thrusters and the momentum exchange devices according to at least a portion of the first and the second control inputs, wherein the controlling includes commanding to the momentum exchange devices to unload the stored momentum and commanding to the thrusters to generate a force and a torque to maintain or change the pose of the spacecraft and to compensate for a torque generated by the momentum exchange devices unloading the stored momentum.
 15. The control system of claim 14, wherein the outer-loop controller uses a model predictive control (MPC) that optimizes a cost function over a receding horizon subject to constraints on the pose of the spacecraft and constraints on inputs to the thrusters, wherein the cost function includes components for controlling the pose of the spacecraft and the momentum stored by the momentum exchange devices, and wherein the inner-loop controller reduces an error between the orientation of the spacecraft and a target orientation of the spacecraft.
 16. The control system of claim 15, wherein the cost function includes a weighted combination of multiple components including a component for a position of the spacecraft penalizing a displacement of the spacecraft from a desired position, a component for an attitude of the spacecraft penalizing larger values of Euler Angles of the spacecraft, a component for the stored momentum penalizing a larger value of a magnitude of the stored momentum, a component for an objective of the operation of the spacecraft, and a component ensuring a stability of the operation of the spacecraft.
 17. The control system of claim 14, wherein the control inputs are determined iteratively, and wherein for at least one iteration, the cost function module updates one or combination of the components of the cost function and weights of the components of the cost function based on a change of a target operation of the spacecraft.
 18. A spacecraft comprising: a set of thrusters for changing a pose of the spacecraft, wherein the thrusters are located on a single face of the spacecraft; a set of momentum exchange devices for absorbing disturbance torques acting on the spacecraft; and the control system of claim 15 for controlling the thrusters and the momentum exchange devices, wherein the constraints on the inputs to the thrusters define a range of rotations of the thrusters forcing thrusts of the thrusters to lie within an interior of a pyramid.
 19. A spacecraft, comprising: a set of thrusters for changing a pose of the spacecraft, wherein the thrusters are located on a single face of the spacecraft; a set of momentum exchange devices for absorbing disturbance torques acting on the spacecraft; and a control system for controlling concurrently operations of the thrusters and the momentum exchange devices, the control system comprising at least one processor for executing modules of the control system, the modules comprising: an inner-loop controller for determining first control inputs to momentum exchange devices for controlling an orientation of the spacecraft; an outer-loop controller for determining second control inputs to thrusters of the spacecraft for concurrently controlling a pose of the spacecraft and a momentum stored by the momentum exchange devices of the spacecraft, wherein the outer-loop controller determines the second control inputs using a model of dynamics of the spacecraft, wherein the model includes dynamics of the inner-loop control, such that the outer-loop controller accounts for effects of actuation of the momentum exchange devices according to the first control inputs determined by the inner-loop controller; and a mapper for controlling the thrusters and the momentum exchange devices according to at least a portion of the first and the second control inputs, wherein the controlling includes commanding to the momentum exchange devices to unload the stored momentum and commanding to the thrusters to generate a force and a torque to maintain or change the pose of the spacecraft and to compensate for a torque generated by the momentum exchange devices unloading the stored momentum.
 20. The spacecraft of claim 14, wherein the outer-loop controller uses a model predictive control (MPC) that optimizes a cost function over a receding horizon subject to constraints on the pose of the spacecraft and constraints on inputs to the thrusters, wherein the cost function includes components for controlling the pose of the spacecraft and the momentum stored by the momentum exchange devices, and wherein the inner-loop controller reduces an error between the orientation of the spacecraft and a target orientation of the spacecraft. 